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As discussed in [452, p. 362] and exemplified in
§C.17.6, to diagonalize a system, we must find the
eigenvectors of
by solving
for
,
, where
is simply the
th pole
(eigenvalue of
). The
eigenvectors
are collected into a
similarity transformation matrix:
If there are coupled repeated poles, the corresponding missing
eigenvectors can be replaced by generalized eigenvectors.2.12 The
matrix is then used to
diagonalize the system by means of a simple change of
coordinates:
The new diagonalized system is then
where
The transformed system describes the same system as in Eq.(1.8)
relative to new state-variable coordinates
. For example,
it can be checked that the transfer-function matrix is unchanged.
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